Introduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measures

PATH - PEPENDENT PARABOLIC PDE S AND
PATH - DEPENDENT F EYNMAN -K AC FORMULA
Jocelyne Bion-Nadal
CNRS, CMAP Ecole Polytechnique
Bachelier Paris, january 8 2016

Dynamic Risk Measures and Path-Dependent second order PDEs, SEFE,
Fred Benth and Giulia Di Nunno Eds, Springer Proceedings inMathematics and Statistics Volume 138, 2016

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Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measuresO UTLINE
1I NTRODUCTION

2PATH DEPENDENT SECOND ORDER PDE S

3M ARTINGALE PROBLEM FOR SECOND ORDER ELLIPTIC
DIFFERENTIAL OPERATORS WITH PATH DEPENDENTCOEFFICIENTS
Martingale problem introduced by Stroock and Varadhan
Path dependent martingale problem
Existence and uniqueness of a solution to the path dependent
martingale problem
Path dependent integro differential operators

4

T IME CONSISTENT DYNAMIC RISK MEASURES

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Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measures

I NTRODUCTION
The field of path dependent PDEs first started in 2010 when Peng asked in[Peng, ICM, 2010] wether a BSDE (Backward Stochastic Differential
Equations) could be considered as a solution to a path dependent PDE.
In line with the recent litterature, a solution to a path dependent second order
PDE
H(u, ω, φ(u, ω), ∂u φ(u, ω), Dx φ(u, ω), D2x φ(u, ω)) = 0(1)
is searched as a progressive function φ(u, ω) ( i.e. a path dependent function
depending at time u on all the path ω up to time u).

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Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measuresCÀDLÀG PATHS
The notion of REGULAR SOLUTION for a path dependent PDE (1) needs todeal with càdlàg paths.
To define partial derivatives Dx φ(u, ω) and D2x φ(u, ω) at (u0 , ω0 ), one needs
to assume that φ(u0 , ω) is defined for paths ω admitting a jump at time u0 .
S. Peng has introduced in [ Peng 2012] a notion of regular and viscosity
solution for a path dependent second order PDE based on the notions ofcontinuity and partial derivatives introduced by Dupire [Dupire 2009].
The main drawback for this approach based on [Dupire 2009] is that theuniform norm topology on the set of càdlàg paths is not separable, it is not a
Polish space.

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Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measures

V ISCOSITY SOLUTION ON CONTINUOUS PATHS
Recently Ekren Keller Touzi and Zhang [ 2014] and also Ren Touzi Zhang[2014] proposed a notion of viscosity solution for path dependent PDEs inthe setting of continuous paths. These works are motivated by the fact that acontinuous function defined on the set of continuous paths does not have aunique extension into a continuous function on the set of càdlàg paths.

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Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measures

N EW APPROACH
In the paper [Dynamic Risk Measures and Path-Dependent second order
PDEs,2015] I introduce a new notion of regular and viscosity solution for
path dependent second order PDEs, making use of the Skorokhod topology
on the set of càdlàg paths. Thus Ω is a Polish space. To define the regularity
properties of a progressive function φ we introduce a one to one
correspondance between progressive functions in 2 variables and strictly
progressive functions in 3 variables.
Our study allows then to define the notion of viscosity solution for pathdependent functions defined only on the set of continuous paths.

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Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measures

C ONSTRUCTION OF S OLUTIONS
Making use of the Martingale Problem Approach for integro differential
operators with path dependent coefficients [J. Bion-Nadal 2015], weconstruct then time-consistent dynamic risk measures on the set Ω of càdlàg
paths. These risk measures provide viscosity solutions for path dependentsemi-linear second order PDEs.
This approach is motivated by the Feynman Kac formula and more
specifically by the link between solutions of parabolic second order PDEs
and probability measures solutions to a martingale problem. The martingale
problem has been first introduced and studied by Stroock and Varadhan(1969) in the case of continuous diffusion processes.

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Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measuresO UTLINE
1I NTRODUCTION

2PATH DEPENDENT SECOND ORDER PDE S

3M ARTINGALE PROBLEM FOR SECOND ORDER ELLIPTIC
DIFFERENTIAL OPERATORS WITH PATH DEPENDENTCOEFFICIENTS
Martingale problem introduced by Stroock and Varadhan
Path dependent martingale problem
Existence and uniqueness of a solution to the path dependent
martingale problem
Path dependent integro differential operators

4

T IME CONSISTENT DYNAMIC RISK MEASURES

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Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measures

T OPOLOGY
In all the following, Ω IS THE SET OF CÀDLÀG PATHS D(IR+ , IRn )ENDOWED WITH THE S KOROKHOD TOPOLOGY
d(ωn , ω) → 0 if there is a sequence λn : IR+ → IR+ strictly increasing,
λn (0) = 0, such that ||Id − λn ||∞ → 0 , and for all K > 0,supt≤K ||ω(t) − ωn ◦ λn (t)|| → 0
T HE SET OF CÀDLÀG PATHS WITH THE S KOROKHOD TOPOLOGY IS AP OLISH SPACE ( metrizable and separable). Polish spaces have niceproperties:
Existence of regular conditional probability distributions
Equivalence between relative compactness and tightness for a set ofprobability measures
The Borel σ-algebra is countably generated.
T HE SET OF CÀDLÀG PATHS WITH THE UNIFORM NORM TOPOLOGY IS
NOT A P OLISH SPACE. It is not separable.

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Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measures

N EW APPROACH FOR PROGRESSIVE FUNCTIONSD EFINITION
Let Y be a metrizable space. A function f : IR+ × Ω → Y is progressive if
f (s, ω) = f (s, ω 0 ) for all ω, ω 0 such that ω|[0,s] = ω 0 |[0,s] .
To every progressive function f : IR+ × Ω → Y we associate a uniquefunction f defined on IR+ × Ω × IRn by
f (s, ω, x) = f (s, ω ∗s x)
ω ∗s x(u)= ω(u) ∀u < sω ∗s x(u)= x ∀s ≤ u(2)0
f is strictly progressive, i.e. f (s, ω, x) = f (s, ω 0 , x) if ω|[0,s[ = ω|[0,s[f → f is a one to one correspondance, f (s, ω) = f (s, ω, Xs (ω)).
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Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measures

R EGULAR SOLUTION OF A PATH DEPENDENT PDE
D EFINITION
A progressive function v on IR+ × Ω is a regular solution to the following
path dependent second order PDE
H(u, ω, v(u, ω), ∂u v(u, ω), Dx v(u, ω), D2x v(u, ω)) = 0(3)

if the function v belongs to C 1,0,2 (IR+ × Ω × IRn ) and if the usual partial
derivatives of v satisfy the equation
H(u, ω ∗u x, v(u, ω, x), ∂u v(u, ω, x), Dx v(u, ω, x), D2x v(u, ω, x) = 0(4)

with v(u, ω, x) = v(u, ω ∗u x)(ω ∗u x)(s) = ω(s) ∀s < u, and (ω ∗u x)(s) = x ∀s ≥ u. The partial
derivatives of v are the usual one, the continuity notion for v is the usual one.
Sufficient to assume that v ∈ C 1,0,2 (X) where X = {(s, ω, x), ω = ω ∗s x}
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Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measures

C ONTINUITY IN VISCOSITY SENSE
D EFINITION
A progressively measurable function v defined on IR+ × Ω is continuous inviscosity sense at (r, ω0 ) ifv(r, ω0 ) = lim {v(s, ω), (s, ω) ∈ D (r, ω0 )}
→0

(5)

where
D (r, ω0 ) ={(s, ω), r ≤ s < r + , ω(u) = ω0 (u), ∀0 ≤ u ≤ rω(u) = ω(s) ∀u ≥ s, and supr≤u≤s ||ω(u) − ω0 (r)|| < } (6)v is lower (resp upper) semi continuous in viscosity sense if equation (5) issatisfied replacing lim by lim inf( resp lim sup).

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Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measures

V ISCOSITY SUPERSOLUTION ON THE SET OF CÀDLÀGPATHS
D EFINITION
Let v be a progressively measurable function on (IR+ × Ω, (Bt )) where Ω isthe set of càdlàg paths with the Skorokhod topology and (Bt ) the canonicalfiltration.
v is a viscosity supersolution of (3) if v is lower semi-continuous in viscosity
sense, and if for all (t0 , ω0 ) ∈ IR+ × Ω, there exists  > 0 such thatv is bounded from below on D (t0 , ω0 ).
for all strictly progressive function φ ∈ Cb1,0,2 (IR+ × Ω × IRn ) such thatv(t0 , ω0 ) = φ(t0 , ω0 , ω0 (t0 )), and (t0 , ω0 ) is a minimizer of v − φ onD (t0 , ω0 ).
H(u, ω ∗u x, φ(u, ω, x), ∂u φ(u, ω, x), Dx φ(u, ω, x), D2x φ(u, ω, x) ≥ 0
at point (t0 , ω0 , ω0 (t0 )).
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Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measures

V ISCOSITY SOLUTION ON CONTINUOUS PATHSD EFINITION
A progressively measurable function v on IR+ × C(IR+ , IRn ) is a viscosity
supersolution of H(u, ω, v(u, ω), ∂u v(u, ω), Dx v(u, ω), D2x v(u, ω)) = 0 if v
satisfies the conditions of the previous theorem replacing D (r, ω0 ) byD̃ (r, ω0 ) where D̃ (r, ω0 ) is the intersection of D (r, ω0 ) with the set ofcontinuous paths

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Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Martingale problem introduced by Stroock and Varadhan
Path dependent second order PDEs
Path dependent martingale problem
Martingale problem for second order elliptic differential operators with path dependentExistencecoefficientsand uniqueness of a solution to the path dependent martingale problem
Time consistent dynamic risk measures
Path dependent integro differential operatorsO UTLINE
1I NTRODUCTION

2PATH DEPENDENT SECOND ORDER PDE S

3M ARTINGALE PROBLEM FOR SECOND ORDER ELLIPTIC
DIFFERENTIAL OPERATORS WITH PATH DEPENDENTCOEFFICIENTS
Martingale problem introduced by Stroock and Varadhan
Path dependent martingale problem
Existence and uniqueness of a solution to the path dependent
martingale problem
Path dependent integro differential operators

4

T IME CONSISTENT DYNAMIC RISK MEASURES
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Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Martingale problem introduced by Stroock and Varadhan
Path dependent second order PDEs
Path dependent martingale problem
Martingale problem for second order elliptic differential operators with path dependentExistencecoefficientsand uniqueness of a solution to the path dependent martingale problem
Time consistent dynamic risk measures
Path dependent integro differential operators

M ARTINGALE PROBLEM OF S TROOCK AND VARADHAN
The martingale problem associated with a second order elliptic differential
operator has been introduced and studied By Stroock and Varadhan["Diffusion processes with continuous coefficients I and II",
Communications on Pure and Applied Mathematics,1969]
Second order elliptic differential operator:
Lta,b =

n

n

i,j=1

i=1

X
∂2
∂
1X
aij (t, x)+bi (t, x)2∂xi ∂xj∂xi

The operator La,b is acting on C0∞ (IRn ) (functions C ∞ with compact support).

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Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Martingale problem introduced by Stroock and Varadhan
Path dependent second order PDEs
Path dependent martingale problem
Martingale problem for second order elliptic differential operators with path dependentExistencecoefficientsand uniqueness of a solution to the path dependent martingale problem
Time consistent dynamic risk measures
Path dependent integro differential operators

M ARTINGALE PROBLEM OF S TROOCK AND VARADHANState space: (C([0, ∞[, IRn ); Xt is the canonical process: Xt (ω) = ω(t)Bt is the σ-algebra generated by (Xu )u≤t .
Let 0 ≤ r and y ∈ IRn . A PROBABILITY MEASURE Q on the space ofcontinuous paths C([0, ∞[, IRn ) IS A SOLUTION TO THE MARTINGALE
PROBLEM FOR La,b starting from y at time r if for all f ∈ C0∞ (IRn ),a,bYr,t= f (Xt ) − f (Xr ) −Z tLua,b (f )(u, Xu )du(7)r

is a Q martingale on (C([0, ∞[, IRn ), Bt ) and if Q({ω(u) = y ∀u ≤ r}) = 1

Lua,b (f )(u, Xu ) =

n

n

1

1

X
1X
∂2f
∂f
aij (u, Xu )bi (u, Xu )(Xu ) +(Xu )2∂xi ∂xj∂xi

17/ 43
Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Martingale problem introduced by Stroock and Varadhan
Path dependent second order PDEs
Path dependent martingale problem
Martingale problem for second order elliptic differential operators with path dependentExistencecoefficientsand uniqueness of a solution to the path dependent martingale problem
Time consistent dynamic risk measures
Path dependent integro differential operators

F EYNMAN K AC FORMULA
Stroock and Varadhan have proved the existence and the uniqueness of thesolution to the martingale problem associated to the operator La,b startingfrom x at time t assuming that a is a continuous bounded function on
IR+ × IRn with values in the set of definite positive matrices and b ismeasurable bounded: Qa,bt,xT HE F EYNMAN K AC FORMULA establishes a link between a solution of aparabolic second order PDE and probability measures solutions to amartingale problem. Under regularity conditions there is a unique solution v
to the PDE: ∂u v(t, x) + La,b v(t, x) = 0 , v(T, .) = h with
La,b v(t, x) = 12 Tr(a(t, x))D2x (v)(t, x) + b(t, x)∗ Dx v(t, x).
From the F EYNMAN K AC FORMULA v(t, x) = EQa,b(h(XT ))t,x(Xu ) is the canonical process.
One natural way to construct and study path dependent parabolic second
order partial differential equations is thus to start with probability measuressolution to the path dependent martingale problem associated to the operator
La,b for path dependent coefficients a and b.
18/ 43
Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Martingale problem introduced by Stroock and Varadhan
Path dependent second order PDEs
Path dependent martingale problem
Martingale problem for second order elliptic differential operators with path dependentExistencecoefficientsand uniqueness of a solution to the path dependent martingale problem
Time consistent dynamic risk measures
Path dependent integro differential operators

PATH DEPENDENT MARTINGALE PROBLEM
I have recently studied the martingale problem associated with an integro
differential operator with path dependent coefficients. We consider here thecase where there is no jump term.
We consider the following path dependent operator:
La,b (t, ω) =

n

n

1

1

X
1X
∂2
∂
aij (t, ω)+bi (t, ω)2∂xi ∂xj∂xi(8)

The functions a and b are defined on IR+ × Ω where Ω is the set of càdlàg
paths. For given t, a(t, ω) and b(t, ω) depend on the whole trajectory of ω up
to time t.

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Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Martingale problem introduced by Stroock and Varadhan
Path dependent second order PDEs
Path dependent martingale problem
Martingale problem for second order elliptic differential operators with path dependentExistencecoefficientsand uniqueness of a solution to the path dependent martingale problem
Time consistent dynamic risk measures
Path dependent integro differential operators

PATH DEPENDENT MARTINGALE PROBLEM
Let Ω = D([0, ∞[, IRn ).
D EFINITION
Let r ≥ 0, ω0 ∈ Ω . A probability measure Q on the space Ω is a solution to
the path dependent martingale problem for La,b (t, ω) starting from ω0 at timer if for all f ∈ C0∞ (IRn ),
a,b,M
Yr,t
= f (Xt ) − f (Xr ) −Z t(La,b (u, ω)(f )(Xu )du(9)r

is a Q martingale on (Ω, Bt ) and if
Q({ω ∈ Ω |ω|[0,r] = ω0 |[0,r] }) = 1

20/ 43
Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Martingale problem introduced by Stroock and Varadhan
Path dependent second order PDEs
Path dependent martingale problem
Martingale problem for second order elliptic differential operators with path dependentExistencecoefficientsand uniqueness of a solution to the path dependent martingale problem
Time consistent dynamic risk measures
Path dependent integro differential operators

PATH DEPENDENT MARTINGALE PROBLEM
T HEOREM
Assume that a and b are bounded. Let Q be a probability measure on Ω suchthat Q({ω ∈ Ω |ω|[0,r] = ω0 |[0,r] }) = 1.
The following properties are equivalent :For all f ∈ C0∞ (IRd ),
a,b,M
Yr,t
(f ) = f (Xt ) − f (Xr ) −Z tLa,b (u, ω)(f )(Xu )du(10)∂+ La,b (u, ω)(f )(u, Xu )du
∂u

(11)

r

is a (Q, Bt ) martingale
a,b,M
For all f ∈ Cb1,2 (IR+ × IRn ), Zr,t(f ) =Z t
f (t, Xt ) − f (r, Xr ) −(ris a (Q, Bt ) martingale.
21/ 43
Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Martingale problem introduced by Stroock and Varadhan
Path dependent second order PDEs
Path dependent martingale problem
Martingale problem for second order elliptic differential operators with path dependentExistencecoefficientsand uniqueness of a solution to the path dependent martingale problem
Time consistent dynamic risk measures
Path dependent integro differential operators

PATH DEPENDENT MARTINGALE PROBLEM
T HEOREM
For all φ ∈ Cb1,0,2 (IR+ × Ω × IRn ) strictly progressive,Z tφ(t, ω, Xt ) − φ(r, ω, Xr ) −[r∂+ La,b (u, ω)]φ(u, ω, Xu (ω))du∂u

is a (Q, Bt ) martingale.
For all g : IR+ × Ω → IR progressive, such that g(g(s, ω, x) = g(s, ω ∗s x)) belongs to Cb1,0,2 (IR+ × Ω × IRn ),Z tg(t, ω) − g(r, ω) −[r∂
+ La,b (u, ω)](g)(u, ω, Xu (ω))du∂u

is a (Q, Bt ) martingale.

22/ 43
Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Martingale problem introduced by Stroock and Varadhan
Path dependent second order PDEs
Path dependent martingale problem
Martingale problem for second order elliptic differential operators with path dependentExistencecoefficientsand uniqueness of a solution to the path dependent martingale problem
Time consistent dynamic risk measures
Path dependent integro differential operators

PATH DEPENDENT MARTINGALE PROBLEM
For φ ∈ Cb1,0,2 (IR+ × Ω × IRn ),La,b (u, ω)(φ)(u, ω, Xu (ω)) =
+

n

n

1

1

X
1X
∂2φ
∂φ
aij (u, ω)(u, ω, Xu (ω)) +bi (u, ω)(u, ω, Xu (ω))
2∂xi ∂xj∂xi

The martingale problem studied by Stroock and Varadhan is a particular caseof the above path dependent martingale problem with a(t, ω) = ã(t, Xt (ω)),
b(t, ω) = b̃(t, Xt (ω)), ã,b̃ defined on IR+ × IRn . W HICH CONTINUITY
ASSUMPTION ON a? Recall that Ω is the set of càdlàg paths.
D EFINITION
A progressive function φ defined on IR+ × Ω is progressively continuous if φ(φ(u, ω, x) = φ(u, ω ∗u x)) is continuous on IR+ × Ω × IRn .
Motivation: If ã is continuous, a given by a(t, ω) = ã(t, Xt (ω)) isprogressively continuous but not continuous on the set of càdlàg paths.
23/ 43
Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Martingale problem introduced by Stroock and Varadhan
Path dependent second order PDEs
Path dependent martingale problem
Martingale problem for second order elliptic differential operators with path dependentExistencecoefficientsand uniqueness of a solution to the path dependent martingale problem
Time consistent dynamic risk measures
Path dependent integro differential operators

E XISTENCE AND UNIQUENESS OF A SOLUTION
T HEOREM
Let a be a progressively continuous bounded function defined on IR+ × Ωwith values in the set of non negative matrices.
Assume that a(s, ω) is invertible for all (s, ω).
Let b be a progressively measurable bounded function defined on IR+ × Ωwith values in IRn .
For all (r, ω0 ), the martingale problem for La,ab starting from ω0 at time r
admits a solution Qa,abr,ω0 on the set of càdlàg paths.
Under a stronger continuity assumption on a (including the Lipschitz case)there is a unique solution to the martingale problem for La,ab starting from
ω0 at time r.
If the function a is δ-delayed which means that a(u, ω) = a(u, ω, Xu (ω))
where a(u, ω, x) depends only on ω up to time u − δ there is also a uniquesolution to the martingale problem for La,ab starting from ω0 at time r.

24/ 43
Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Martingale problem introduced by Stroock and Varadhan
Path dependent second order PDEs
Path dependent martingale problem
Martingale problem for second order elliptic differential operators with path dependentExistencecoefficientsand uniqueness of a solution to the path dependent martingale problem
Time consistent dynamic risk measures
Path dependent integro differential operators

T HE ROLE OF CONTINUOUS PATHSP ROPOSITIONa,ab
Every probability measure Qa,abr,ω0 solution to the martingale problem for L
starting from ω0 at time r is supported by paths which are continuous after
time r, i.e.continuous on [r, ∞[.
More preciselynQa,abr,ω0 ({ω, ω(u) = ω0 (u) ∀u ≤ r, and ω|[r,∞[ ∈ C([r, ∞[, IR )} = 1

C OROLLARY
For all continuous path ω0 and all r, the support of the probability measureQa,abr,ω0 is contained in the set of continuous paths:
a,b
Qr,ω
C([IR+ , IRn )) = 1
0
25/ 43
Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Martingale problem introduced by Stroock and Varadhan
Path dependent second order PDEs
Path dependent martingale problem
Martingale problem for second order elliptic differential operators with path dependentExistencecoefficientsand uniqueness of a solution to the path dependent martingale problem
Time consistent dynamic risk measures
Path dependent integro differential operators

F ELLER PROPERTYT HEOREM
Assume that a and b are progressively continuous bounded. Assume that
there is a unique solution to the martingale problem for La,0 starting from
ω0 at time r. Then there is a unique solution to the martingale problem for
La,ab starting from ω0 at time r. Furthermore the map(r, ω, x) ∈ IR+ × Ω × IRn → Qa,abr,ω∗r x ∈ M1 (Ω)
is continuous on {(r, ω, x) | ω = ω ∗r x}
The set of probability measures M1 (Ω) is endowed with the weak topology.
Let h(ω) = h(ω, ω(T)), h continuous, h(ω, x) = h(ω 0 , x) if
ω(u) = ω 0 (u), ∀u < T.
P ROPOSITION
Let v(r, ω) = Qa,abr,ω (h). v is continuous on X. The function v is a viscosity
solution of ∂t v(t, ω) + La,ab v(t, ω) = 0 , v(T, ω) = h(ω)
Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formula

26/ 43

Introduction
Martingale problem introduced by Stroock and Varadhan
Path dependent second order PDEs
Path dependent martingale problem
Martingale problem for second order elliptic differential operators with path dependentExistencecoefficientsand uniqueness of a solution to the path dependent martingale problem
Time consistent dynamic risk measures
Path dependent integro differential operators

PATH DEPENDENT INTEGRO DIFFERENTIAL OPERATORS
Consider the following path dependent operators
La,b (t, ω) =

n

n

1

1

X
1X
∂2
∂
aij (t, ω)+bi (t, ω)2∂xi ∂xj
∂xi

(12)

and
M

Z
[f (x + y) − f (x) −K (t, ω)(f )(x) =R
I n −{0}y∗ ∇f (x)]M(t, ω, dy)
1 + ||y||2(13)

a, b, M are progressive functions.

27/ 43
Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Martingale problem introduced by Stroock and Varadhan
Path dependent second order PDEs
Path dependent martingale problem
Martingale problem for second order elliptic differential operators with path dependentExistencecoefficientsand uniqueness of a solution to the path dependent martingale problem
Time consistent dynamic risk measures
Path dependent integro differential operators

PATH DEPENDENT MARTINGALE PROBLEM
Let Ω = D([0, ∞[, IRn ).
D EFINITION
Let r ≥ 0, ω0 ∈ Ω . A probability measure Q on the space Ω is a solution to
the path dependent martingale problem for La,b (t, ω) + K M (t, ω) startingfrom ω0 at time r if for all f ∈ C0∞ (IRn ),
a,b,M
Yr,t
= f (Xt ) − f (Xr ) −Z t(La,b (u, ω) + K M (u, ω))(f )(Xu )du

(14)

r

is a Q martingale on (Ω, Bt ) and if
Q({ω ∈ Ω |ω|[0,r] = ω0 |[0,r] }) = 1

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Path-pependent PDEs and Feynman-Kac formulaIntroduction
Martingale problem introduced by Stroock and Varadhan
Path dependent second order PDEs
Path dependent martingale problem
Martingale problem for second order elliptic differential operators with path dependentExistencecoefficientsand uniqueness of a solution to the path dependent martingale problem
Time consistent dynamic risk measures
Path dependent integro differential operators(A)- BOUNDEDNESS CONDITIOND EFINITION
The functions a, b, M satisfy the (A)-boundedness condition if
sup||a(s, ω)|| ≤ A

s≥0,ω∈Ω

sup||b(s, ω)|| ≤ A

s≥0,ω∈Ω||y||2M(s, ω, dy) ≤ A
2R
I n −{0} 1 + ||y||Z
sups≥0,ω∈Ω

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Path-pependent PDEs and Feynman-Kac formulaIntroduction
Martingale problem introduced by Stroock and Varadhan
Path dependent second order PDEs
Path dependent martingale problem
Martingale problem for second order elliptic differential operators with path dependentExistencecoefficientsand uniqueness of a solution to the path dependent martingale problem
Time consistent dynamic risk measures
Path dependent integro differential operators

E XISTENCE AND UNIQUENESST HEOREM
Assume that(a, b, M) satisfy the A-boundedness condtion∀s, ω, a(s, ω) is positive definite
a is progressively continuous (i.e. (s, ω, x) → a(s, ω, x) = a(s, ω ∗s x) is
continuous)
R||y||2∀φ, (s, ω, x) → RI d −{0} 1+||y||2 φ(y)M(s, ω ∗s x, dy) is continuous.
Then for all (r, ω0 ) there exists a solution Qr,ω0 to the path dependent
martingale problem for La,b (t, ω) + K M (t, ω) starting from ω0 at time r, i.e.
such thatQr,ω0 ({ω| ω|[0,r] = ω0 |[0,r] }) = 1
Under more restrictive assumptions the solution is unique.
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Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measuresO UTLINE
1I NTRODUCTION

2PATH DEPENDENT SECOND ORDER PDE S

3M ARTINGALE PROBLEM FOR SECOND ORDER ELLIPTIC
DIFFERENTIAL OPERATORS WITH PATH DEPENDENTCOEFFICIENTS
Martingale problem introduced by Stroock and Varadhan
Path dependent martingale problem
Existence and uniqueness of a solution to the path dependent
martingale problem
Path dependent integro differential operators

4

T IME CONSISTENT DYNAMIC RISK MEASURES
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Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measures

T IME CONSISTENT DYNAMIC RISK MEASURES
Recall the following way of constructing time consistent dynamic risk
measures [J.Bion-Nadal 2008].
P ROPOSITION
Given a stable set Q of probability measures all equivalent to Q0 and apenalty (αs,t ) defined on Q satisfying the local property and the cocycle
condition,ρst (X) = esssupQ∈Q (EQ (X|Fs ) − αst (Q))
defines a time consistent dynamic risk measure on L∞ (Ω, B, (Bt ), Q0 ) or on
Lp (Ω, B, (Bt ), Q0 ) if the corresponding integrability conditions are satisfied.
That is: ρst : Lp (Ω, Bt , Q0 ) → Lp (Ω, Bs , Q0 ), ρst is convex, translation
invariant by elements of Lp (Ω, Bs , Q0 ) and ρr,t (X) = ρr,s (ρs,t (X)) for all
X ∈ Lp (Ω, Bt , Q0 ) and r ≤ s ≤ t.

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Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measures

S TABLE SET OF PROPBABILITY MEASURESD EFINITIONA set Q of equivalent probability measures on a filtered probability space(Ω, B, (Bt )) is stable if it satisfies the two following properties:
1

Stability by composition
For all s ≥ 0 for all Q and R in Q, there is a probability measure S in Q
such that for all X bounded B-measurable,
ES (X) = EQ (ER (X|Bs ))2

Stability by bifurcation
For all s ≥ 0, for all Q and R in Q, for all A ∈ Bs , there is a probability
measure S in Q such that for all X bounded B-measurable,
ES (X|Bs ) = 1A EQ (X|Bs ) + 1Ac ER (X|Bs )33/ 43
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Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measures

P ENALTIESD EFINITION
A penalty function α defined on a stable set Q of probability measures allequivalent is a family of maps (αs,t ), s ≤ t , defined on Q with values in theset of Bs -measurable maps.
I ) It is local:

if for all Q, R in Q, for all s, for all A in Bs , the assertion
1A EQ (X|Bs ) = 1A ER (X|Bs ) for all X bounded Bt measurable implies
that 1A αs,t (Q) = 1A αs,t (R).
II ) It satisfies the cocycle condition if for all r ≤ s ≤ t, for all Q in Q,

αr,t (Q) = αr,s (Q) + EQ (αs,t (Q)|Fr )34/ 43
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Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measures

M ULTIVALUED MAPPING
For all (s, ω, x) consider Λ(s, ω, x) ⊂ IRn such that Λ(s, ω, x) = Λ(s, ω 0 , x) if
ω(u) = ω 0 (u) ∀u ≤ s. Λ is a multivalued mapping on X (the quotient of
IR+ × Ω × IRn by the equivalence relation ∼: (t, ω, x) ∼ (t0 , ω 0 , x0 ) if t = t0 ,x = x0 and ω(u) = ω 0 (u) ∀u < t).
A selector from Λ is a map s such that s(t, ω, x) ∈ Λ(t, ω, x) for all (t, ω, x).
D EFINITION
A multivalued mapping Λ from X into IRn is lower hemicontinuous if
for every closed subset F of IRn , {(t, ω, x) ∈ X : Λ(t, ω, x) ⊂ F} is closed.
Recall the following Michael selection Theorem:
A lower hemicontinuous mapping from a metrizable space into a Banach
space with non empty closed convex values admits a continuous selector.

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Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measures

S TABLE SET OF PROBABILITY MEASURES SOLUTION TO
A MARTINGALE PROBLEM
In all the following a is progressively continuous bounded on IR+ × Ωa(s, ω) invertible for all (s, ω) . For all (r, ω) there is a unique solution Qa,0r,ωto the martingale problem for La,0 starting from ω at time r.
D EFINITION
Let Λ be a closed convex lower hemicontinuous multivalued mapping. Let
L(Λ) be the set of continuous bounded selectors from Λ.
Let r ≥ 0 and ω ∈ Ω. The set Qr,ω (Λ) is the stable set of probabilitya,aλmeasures generated by the probability measures Qr,ω, λ ∈ L(Λ) with000λ(t, ω ) = λ(t, ω , Xt (ω ))Let P be the predictable σ-algebra. Every probability measure in Qr,ω (Λ) isa,aµ
the unique solution Qr,ωto the martingale problem for La,aµ starting from ωat time r for some process µ defined on IR+ × Ω × IRn
P × B(IRn )-measurable Λ valued (µ(u, ω, x) ∈ Λ(u, ω, x)).
Every probability measure in Qr,ω (Λ) is equivalent with Qar,ω = Qa,0r,ω .
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Path-pependent PDEs and Feynman-Kac formula

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Introduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measures

PENALTIESFor 0 ≤ r ≤ s ≤ t, define the penalty αs,t (Qa,aµr,ω ) as follows
a,aµ
a,aµ (
αs,t (Qr,ω
) = EQr,ω

Ztg(u, ω, µ(u, ω))du|Bs )

(15)

s

where g is P × B(IRn )-measurable.

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Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measures

G ROWTH CONDITIONSD EFINITION
1

g satisfies the growth condition (GC1) if there is K > 0, m ∈ IN ∗ and > 0 such that∀y ∈ Λ(u, ω, Xu (ω)), |g(u, ω, y)| ≤ K(1 + sup ||Xs (ω)||)m (1 + ||y||2− )
s≤u

(16)
2

g satisfies the growth condition (GC2) if there is K > 0 such that∀y ∈ Λ(u, ω, Xu (ω)), |g(u, ω, y)| ≤ K(1 + ||y||2 )(17)

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Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measures

BMO CONDITIOND EFINITION
Let C > 0. Let Q be a probability measure.
A progressively measurable process µ belongs to BMO(Q) and has a
BMO norm less or equal to C if for all stopping times τ ,Z ∞EQ (||µs ||2 ds|Fτ ) ≤ C
τThe multivalued mapping Λ is BMO(Q) if there is a map φ ∈ BMO(Q)such that∀(u, ω), sup{||y||, y ∈ Λ(u, ω)} ≤ φ(u, ω)

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Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measures

T IME CONSISTENT DYNAMIC RISK MEASURE ON Lp
T HEOREM
Let (r, ω). Assume that the multivalued set Λ is BMO(Qar,ω ). Let
Q = Qr,ω (Λ) Let r ≤ s ≤ t.
a,aµa,aµ (Y|B ) − αρr,ω
(EQr,ωss,t (Qr,ω ))s,t (Y) = esssupQa,aµr,ω ∈QRt
a,aµwith αs,t (Qr,ω
) = EQa,aµ( g(u, ω, µ(u, ω))du|Bs )r,ωs

Assume that g satisfies the growth condition (GC1). Then (ρr,ωs,t ) defines
a time consistent dynamic risk measure on Lp (Qar,ω , (Bt )) for all
q0 ≤ p < ∞.
Assume that g satisfies the growth condition (GC2). Then (ρr,ωs,t ) defines
a time consistent dynamic risk measure on Lp (Qar,ω , (Bt )) for all
q0 ≤ p ≤ ∞q0 is linked to the BMO norm of the majorant of Λ.
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Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measures

F ELLER PROPERTY FOR THE DYNAMIC RISK MEASURE
D EFINITION
The function h : Ω → IR belongs to Ct if there is a h̃ : Ω × IRn → IR such thath(ω) = h̃(ω, Xt (ω))
h̃(ω, x) = h̃(ω 0 , x) if ω(u) = ω 0 (u) ∀u < tand such that h̃ is continuous bounded on {(ω, x), ω = ω ∗t x} ⊂ Ω × IRn
T HEOREM
Under the same hypothesis. For every function h ∈ Ct , there is a progressive
map R(h) on IR+ × Ω, R(h)(t, ω) = h(ω), such that R(h) is lower semi
continuous on {(u, ω, x), ω = ω ∗u x, u ≤ t}.
0s,ω∀s ∈ [r, t], ∀ω 0 ∈ Ω, ρs,t(h) = R(h)(s, ω 0 )000a∀0 ≤ r ≤ s ≤ t, ρr,ωs,t (h)(ω ) = R(h)(s, ω , ω (s)) Qr,ω a.s.

(18)
(19)
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Jocelyne Bion-Nadal

Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measures

V ISCOSITY SOLUTION
T HEOREM
Assume furthermore that g is upper semicontinuous on{(s, ω, y), (s, ω) ∈ X, y ∈ Λ(s, ω)}. Let h ∈ Ct . The function R(h) is aviscosity supersolution of the path dependent second order PDE−∂u v(u, ω) − Lv(u, ω) − f (u, ω, a(u, ω)Dx v(u, ω))v(t, ω)Lv(u, ω)=f (u, ω, z)

=

= 0
=

f (ω)

1
Tr(a(u, ω)D2x (v)(u, ω))
2
sup (z∗ y − g(u, ω, y))y∈Λ(u,ω)at each point (t0 , ω0 ) such that f (t0 , ω0 , a(t0 , ω0 )z) is finite for all z.
0

0
ρs,ω
s,t (h) = R(h)(s, ω )42/ 43
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Path-pependent PDEs and Feynman-Kac formulaIntroduction
Path dependent second order PDEs
Martingale problem for second order elliptic differential operators with path dependent coefficients
Time consistent dynamic risk measures

V ISCOSITY SOLUTION
T HEOREM
Assume furthermore that Λ is uniformly BMO with respect to a. Assume thatf is progressively continuous. Let h ∈ Ct . The upper semi-continuous
envelop of R(h) in viscosity senseR(h)∗ (s, ω) = lim sup{R(h)(s0 , ω 0 ), (s0 , ω 0 ) ∈ Dη (s, ω)}η→0is a viscosity subsolution of the above path dependent second order PDE.

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Path-pependent PDEs and Feynman-Kac formula